Word or expression for when a person not under pressure is able to see an issue easier than the person under pressure?

There is the phrase: a spectator sees more of the game

Trying to find the origin of that is proving particularly difficult.

The Chinese have a similar phrase: The person on the spot is baffled, the onlooker sees clear. I expect it's a lot more snappy in Mandarin.

The meanings of both are quite clear I think and, I think, apt.

I have an example which is related to but not exactly what you are requesting.

Cooler heads prevail.

People who are calm and focused are the most likely to succeed or to have the greatest influence on events.

wiktionary.org

the ideas or influence of less emotional people prevail. (Used of a tense situation.)

thefreedictionary.com

This phrase suggests that those that are more impervious to pressure will make shrewder decisions, which is not far off your question.